For an algorithm to be suitable for an embedded implementation, it needs to involve only simple steps. This deems methods of the forward-backward-splitting (FBS) type [1]}, such as the proximal gradient method [2]}, appealing candidates. FBS-type algorithms can be used to solve nonlinear optimal control problems with simple input constraints via first eliminating the state sequence and expressing the cost as a function of the sequence of inputs alone — the so-called single shooting formulation. However, despite its simplicity, FBS, like all first-order methods, can exhibit slow convergence. Its convergence rate is at best Q-linear with a Q-factor close to one for ill-conditioned problems such as most nonlinear MPC problems.
